Rigging default alternatives
It's not really a different style of armature.  It's just an armature without meaningfully placed bone tails.
The essence of a bone, for skeletal animation purposes, lies in it having two characteristics:

An origin about which it can rotate (and/or scale; translation doesn't really need an origin);
A hierarchical structure from which it can inherit a transformation from its parents.

To do those two things, you don't need bone tails.  And in fact, many skeletal deformation systems have no conception of bone tails-- their bones are points, not lines.  The only thing you need for a bone is the head of the bone.  This is common in game engines, where any unnecessary data is scrubbed out in the name of efficiency, and that's why you're going to see game rip models without meaningful bone tails.  The game never stored their tails, because it didn't need them.
That doesn't mean that meaningfully placed bone tails aren't useful in Blender.  In Blender, bone tails do three things:

They establish bone length data for autoweights and bone envelopes;
They, along with bone roll, establish per-bone local axes;
They act as an extra pocket of information that you can use with bone constraints.

All three of those things are very useful.  Autoweights are a great start on weights that save a ton of time.  It is a pain to fix an A-pose arm twist bone when its local axes are oriented with the world axes at rest.  And good luck creating even a simple leg IK without a meaningfully placed shin bone tail.
However, if you want, you can work with an armature like this.  You can weight paint manually.  You can create non-deforming control bones to deal with constraints, or just not use any constraints.  You can pose it entirely visually, never limiting rotations to any local bone axes.
It's just, you lose out on some easier ways of doing things by ignoring your bone tails.